Lesson 1Complex numbers: algebra, polar form, Euler's formula, roots and basic complex equationsWe go over complex numbers as extensions of real numbers. Students handle algebraic operations, polar and exponential forms, Euler’s formula, roots of complex numbers, and simple complex equations for oscillatory systems.
Algebra of complex numbersModulus, argument, and conjugatePolar and exponential formsEuler’s formula and rotationsRoots and basic complex equationsLesson 2Functions and their properties: polynomial, rational, exponential, logarithmic, and piecewise definitionsThis part reviews main function types used in modelling. We look at polynomial, rational, exponential, logarithmic, and piecewise functions, focusing on domains, ranges, graphs, transformations, and inverses.
Domain and range analysisPolynomial and rational graphsExponential growth and decayLogarithmic functions and inversesPiecewise and step functionsLesson 3Probability and statistics basics: probability rules, discrete and continuous distributions, expected value, variance, combinatorics basicsWe cover probability and statistics tools for engineering data. Students learn probability rules, counting combinations, discrete and continuous distributions, expected value, variance, and basic stats summaries.
Sample spaces and eventsAddition and multiplication rulesCombinatorics and counting methodsDiscrete and continuous variablesExpectation, variance, and spreadLesson 4Sequences and series: convergence tests, Taylor and Maclaurin series, power series representation and radius of convergenceThis part covers sequences and series, focusing on convergence. We test series with standard methods, build power series, do Taylor and Maclaurin expansions, and find radius and interval of convergence.
Limits of sequences and behaviorSeries convergence conceptsComparison and ratio testsPower series and convergence radiusTaylor and Maclaurin seriesLesson 5Limits and continuity: limit laws, indeterminate forms, L'Hôpital's rule, limits at infinityWe cover limits and continuity for proper calculus. Students use limit laws, one-sided limits, indeterminate forms, L’Hôpital’s rule, and limits at infinity for asymptotic function behaviour.
Limit laws and computationsOne-sided limits and continuityRemovable and jump discontinuitiesIndeterminate forms and algebraic tricksL’Hôpital’s rule and limits at infinityLesson 6Applications of integrals: area, volume by revolution, work, accumulation problems, average valueWe see how definite integrals model built-up quantities in engineering. Topics include areas, revolution volumes, variable force work, averages, and real problem integrals.
Area between curves and axesVolumes by disks and washersShell method for volumesWork by variable forcesAverage value of a functionLesson 7Euclidean geometry and trigonometry: triangle properties, circle theorems, trigonometric identities, solving trig equationsThis part revisits geometry and trig for exams. We cover triangle congruence, circle theorems, radians, trig identities, inverse trig, and solving trig equations.
Triangle congruence and similarityCircle theorems and chordsRadian measure and arc lengthCore trigonometric identitiesSolving trigonometric equationsLesson 8Applications of derivatives: optimization, curve sketching, related rates, linearization and approximationsWe use derivatives to analyse and approximate functions. Topics include one-variable optimisation, curve sketching with derivatives, related rates, linearisation, and approximations.
Critical points and extremaFirst and second derivative testsCurve sketching strategiesRelated rates word problemsLinearization and differentialsLesson 9Differential calculus: derivative rules, implicit differentiation, higher-order derivatives, mean value theoremWe build differential calculus for rates of change. Students learn derivative rules, chain and implicit differentiation, higher derivatives, Mean Value Theorem, with focus on skills and meanings.
Limit definition of derivativeBasic derivative rulesChain rule applicationsImplicit differentiation methodsHigher derivatives and MVTLesson 10Linear algebra essentials: systems of linear equations, matrices, determinants, eigenvalues (basic concepts relevant to modelling)This part introduces linear algebra for modelling. We solve linear systems, handle matrices, determinants, and interpret eigenvalues and eigenvectors in simple models for mechanics, electrics, and populations.
Gaussian elimination methodsMatrix operations and inversesDeterminants and Cramer’s ruleEigenvalues and eigenvectors basicsLinear models and applicationsLesson 11Integral calculus: antiderivatives, definite integrals, Fundamental Theorem of Calculus, substitution and integration by partsThis focuses on antiderivatives and definite integrals. We use Fundamental Theorem of Calculus, substitution, parts, and see integrals as signed areas and accumulated change.
Antiderivatives and familiesDefinite integrals as areaFundamental Theorem of CalculusSubstitution and change of variableIntegration by parts strategiesLesson 12Vectors and analytic geometry: vector operations, dot and cross product, lines and planes in 3D, coordinate transformationsThis builds 3D analytic geometry with vectors. We do vector operations, dot and cross products, line and plane equations, distances, projections, and basic coordinate changes.
Vector addition and scalar multiplicationDot product and projectionsCross product and geometryLines and planes in 3D spaceCoordinate changes and rotations
